A small remote village receives radio broadcasts from two radio stations, a news station and a music station. Of the listeners who are tuned to the news station, 70% will remain listening to the news after the station break that occurs each half hour, while 30% will switch to the music station at the station break. Of the listeners who are tuned to the music station, 60% will switch to the news station at the station break, while 40% will remain listening to the music. Suppose everyone is listening to the news at 8:15 A.M. a. Give the stochastic matrix that describes how the radio listeners tend to change stations at each station break. Label the rows and columns. b. Give the initial state vector c. What percentage of the listeners will be listening to the music station at 9:25 A.M. (after the station breaks at 8:30 and 9:00 A.M.)?

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Asked Dec 14, 2019
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A small remote village receives radio broadcasts from two radio stations, a news station and a music station. Of the listeners who are tuned to the news station, 70% will remain listening to the news after the station break that occurs each half hour, while 30% will switch to the music station at the station break. Of the listeners who are tuned to the music station, 60% will switch to the news station at the station break, while 40% will remain listening to the music. Suppose everyone is listening to the news at 8:15 A.M. a. Give the stochastic matrix that describes how the radio listeners tend to change stations at each station break. Label the rows and columns. b. Give the initial state vector c. What percentage of the listeners will be listening to the music station at 9:25 A.M. (after the station breaks at 8:30 and 9:00 A.M.)? 

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Expert Answer

Step 1

Given:

Of the listeners, 70% will remain listening to the news after the station break that occurs each half hour, while 30% will switch to the music station at the station break. where 30% will switch to the music station at the station break. Of the listeners who are tuned to the music station, 60% will switch to the news station at the station break, while 40% will remain listening to the music. And also, it is supposed that everyone is listening to the news at 8:15 A.M.

Step 2

Now for stochastic matrix:

A stochastic matrix is a square matrix whose columns are probability vectors.

Hence in this case:

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The probability vectors are: v; = [0.7,0.3] v; =[0.6,0.4] Therefore the stochastic matrix M is given by: T0.7 0.6 M = 0.3 0.4 Where columns are probability vectors. In which 1st column is for news, 2nd column is for music listeners. In same manner 1st row is for news, 2nd row is for music listeners.

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Step 3

Now for initial vector:

Since from the given information everyone started listening news at ...

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The initial vector is given by: X, =

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Math

Calculus